Maxima of independent sums in the presence of heavy tails

Let
$$Y_{mn}=\max_{1\le i\le m}\sum_{j=1}^n X_{ij},\qquad m,n\ge 1,$$
be a family of extremes, where $X_{ij}$, $i,j\ge 1$, are independent with common subexponential distribution $F$. The limit behavior of $Y_{mn}$ is investigated as $m,n\to\infty$. Various nondegenerate limit laws are obtained (Fréchet and Gumbel), depending on the relative rate of growth of $m,n$ and the tail behavior of $F$.